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LES predictions of mixing enhancement for Jets In
Cross-Flows
C Priere†, LYM Gicquel† §, P Kaufmann‡,W Krebs‡ and T Poinsot]
† European Center For Research and Formation in Advanced Computations, 42
Avenue Coriolis, 31057 Toulouse Cedex 01, France
‡ Siemens PG, 45466 Muhlheim an der Ruhr, Germany
] Institut de Mecanique des Fluides, Allee du Professeur Camille SOULA, 31400
TOULOUSE, France
Abstract. The efficiency of a set of mixing enhancement devices on an array of Jets
In Cross-Flow (JICF) is studied using Large Eddy Simulation (LES). The baseline flow
is a rectangular channel flow on which five JICF’s are installed on each wall (upper
and lower). Mixing devices are fixed tabs installed upstream of the jets. Instantaneous
analyses of the LES fields reveal two large vortical structures developing downstream
of the mixing device. These structures strongly enhance mixing. Comparisons of
the statistically averaged LES predictions against experimental results validate the
predictions. The mixing devices provide a better spatial and temporal homogeneity of
the gas mixture at the exit of the main duct. Even though full temporal and spatial
homogeneity of the gas mixture prior to combustion is not guaranteed with this design,
the probability of finding strong inhomogeneous zones is reduced. More generally, this
study confirms the power of LES to help design actuating devices for flow and mixing
control.
PACS numbers: 00.00, 20.00, 42.10
CERFACS number: TR/CFD/03/11
§ To whom correspondence should be addressed.
LES of mixing enhancement 2
1. Introduction
Upcoming environmental constraints require considerable investments for the design of
the next generation of gas turbine combustion chambers. These research efforts are
aimed to yield low pollutant emissions and fuel efficient designs in agreement with the
new industrial context. Fuel lean combustion is one of the most promising candidates
to meet the requirements. However such designs are often subject to combustion
instabilities [1, 2, 3, 4] involving coupling between combustion and acoustics. If acoustic
resonance were to occur the mechanical constraints imposed on the turbine can result
in mechanical failures. That last scenario and its disastrous consequences are to be
avoided if possible during the design phases. The triggering mechanisms for combustion
instabilities are: periodic formation of inhomogeneous fuel pockets, periodic shedding of
large scale structures and natural amplification of the acoustic waves by the flame front
[5, 6, 7, 8, 9] etc. Real life combustors involve all of the above and control of these sources
still remains a challenge. Experimental benchmarks are necessary to investigate the
design choices as well as the global behavior of the new turbine. Resulting development
costs are heavy and potential design solutions are usually disregarded due to budgetary
restrictions. The advent of the computer simulations offers an interesting approach to
test many options prior to the final design.
Several computational methods have reached maturity to be used in the design
chain. The first approach developed in the forties is based on the Reynolds Averaged
Navier-Stokes (RANS) equations [10]. The method is restricted (in theory) to steady
turbulent reacting flows and requires serious modelling efforts to take into account
combustion instabilities. The second approach, named Large Eddy Simulation (LES)
[11, 12, 13, 14, 15], appears more attractive. In LES, large scale phenomena are naturally
embedded in the governing equations and only the small scales are modelled. This work
adopts the second method to predict fuel mixing enhancement prior to its injection
in the combustion chamber. Two design options defined by Siemens PG, Germany,
are simulated with LES. The first configuration consists of ten opposed in-line Jets
In Cross-Flow (JICF) issuing in a rectangular duct. The second design consists of
the first configuration plus mixing devices located in the rectangular duct, upstream
of the injection plane. LES predictions for both geometries are compared to identify
the potential of the approach for industrial applications. Predictions of the spatial
and temporal variations of fuel concentration with regard to combustion instabilities
constitutes the objective of the work. Homogenisation is clearly improved when the
mixing devices are present. Statistical analyses such as the Probability density functions
(Pdf) and mixing indices illustrate the changes in the mixing mechanisms implied by
the changes in geometry.
The presentation of the LES methodology and the computer code are briefly
given in the first section. The second section specifies the configuration, the grid
and boundary conditions employed in LES. Results are then exposed in two steps. A
detailed description of the flow topologies and the consequences on mixture homogeneity
LES of mixing enhancement 3
are given in section 4.1. The statistical investigation of the LES is then presented
(section 4.2). The precision of the LES results is evaluated at this occasion by a
comparison against experimental measurements of the University of Bochum [16] and
unsteady RANS predictions performed by Siemens PG [17].
2. The LES approach
The theoretical aspects of the LES method and the closures employed for this work are
presented in this section. The description does not intend to be comprehensive and for
further information the reader is pointed to [15]. The numerical implementation of LES
in the computer code AVBP is also given; further information about AVBP is found in
[18].
2.1. The governing equations
LES involves the spatial filtering operation [19]:
f(x, t) =
∫ +∞
−∞
f(x′, t) G(x′,x) dx′, (1)
where G denotes the filter function and f(x, t), is the filtered value of the variable f(x, t).
We consider spatially and temporally invariant and localized filter functions [19], thus
G(x′,x) ≡ G(x′ − x) with properties [19, 20], G(x) = G(−x) and∫ +∞
−∞G(x) dx = 1.
In the mathematical description of compressible turbulent flows with transport of
fuel (as a passive scalar), the primary variables are the density, ρ(x, t), the velocity
vector, ui(x, t), the total energy, E(x, t) ≡ es + 1/2 uiui, and the fuel mass fraction,
Yf(x, t). The application of the filtering operation to the instantaneous transport
equations yields [9]:
∂ρ
∂t+
∂
∂xi
(ρ ui) = 0,
∂
∂t(ρ uj) +
∂
∂xi
(ρ ui uj) = − ∂p
∂xj
+∂τ jk
∂xk
− ∂
∂xi
(ρ Tij),
∂
∂t(ρ E) +
∂
∂xi
(ρ ui E) = −∂qj
∂xj
+∂
∂xj
[(τij − p δij) ui]−∂
∂xj
(ρ Qj) (2)
− ∂
∂xj
(ρ Tijui),
∂
∂t(ρ Yf) +
∂
∂xi(ρ ui Yf) = −∂J i
∂xi− ∂
∂xi(ρ Fi).
In (2), one uses the Favre filtered variable [21], f = ρ f/ρ. The fluid follows the ideal
gas law, p = ρRT and es = Cv T , where T stands for the temperature. The tensor
of viscosity, the heat diffusion vector and the molecular transport of the passive scalar
read respectively:
τij = µ
(∂ui
∂xj+
∂uj
∂xi
)− 2
3
∂uk
∂xkδij,
LES of mixing enhancement 4
qi = −λ∂T
∂xi
, (3)
Ji = −Df
∂Yf
∂xi
.
In (3), µ is the fluid viscosity following Sutherland’s law, λ the heat diffusion coefficient
following Fourier’s law, and Df the fuel diffusion coefficient following Fick’s law.
Variations of the molecular coefficients resulting from the unresolved fluctuations are
neglected hereinafter so that the various expressions for the molecular coefficients
become only function of the filtered field.
The objective of LES is to compute the largest structures of the flow (these
structures are typically larger than the computational mesh size), while the effects of
the smaller scales are modelled. This scale separation is obtained through the filtering
operation, (2), and the unknowns, Tij, Qi, Fi, correspond to the so-called Sub-Grid Scale
(SGS) (cf Lesieur [22] , Sagaut [15] , Ferziger [23]). The unresolved SGS stress tensor,
Tij, require a sub-grid turbulence model. Introducing the concept of SGS turbulent
viscosity most models read (Smagorinsky [24]):
Tij = (uiuj − ui uj) = −2νtSij −1
3Tllδij, (4)
with,
Sij =1
2
(∂ui
∂xj+
∂uj
∂xi
)− 2
3
∂uk
∂xkδij. (5)
In equations (4) & (5), Sij is the resolved strain tensor, νt is the SGS turbulent viscosity.
The aim of the model is to determine νt. Dimensional analysis yields νt ∝ lSGS×√
qSGS
where lSGS is the length scale of the unresolved motion and√
qSGS its velocity scale.
The WALE model [25] (Wall Adapting Local Eddy-viscosity) is used. The expression
for νt then follows:
νt = (Cw4)2(sd
ijsdij)
3/2
(SijSij)5/2+(sdijs
dij)
5/4, (6)
with,
sdij =
1
2(g2
ij + g2ji)−
1
3δij. (7)
In (6), 4, denotes the filter characteristic length and is approximated by the cubic-root
of the cell volume, Cw is the model constant (Cw=0.55), and gij the resolved velocity
gradient.
The SGS energy flux, Qi = Cp(T ui−T ui), is modelled by use of the eddy diffusivity
concept with a turbulent Prandtl number, Prt = 0.9, so that κt = νt Cp/Prt and:
Qi = −κt∂T
∂xi
. (8)
Note that T is the modified filtered temperature and satisfies the modified filtered state
equation, p = ρ R T [26, 27, 28, 29].
LES of mixing enhancement 5
The SGS flux, Fi = (uiYf − ui Yf), is modelled by:
Fi = −Dt∂Yf
∂xi
. (9)
In (9), Dt = νt/Sct, denotes the fuel SGS turbulent diffusivity where, Sct is the turbulent
Schmidt number (Sct = 0.7). Although the performances of the closures could be
improved through the use of a dynamic formulation (Germano [30] , Moin [31] , Lilly
[32], Meneveau [33], Ghosal [34] ...), they are sufficient to investigate the effects of the
mixing devices.
2.2. General description of the code
The LES code (AVBP) solves the LES transport equations (cf. section 2.1) on
structured, unstructured or hybrid grids (cf. http://www.cerfacs.fr). The numerical
approach is based on finite-volume schemes using the cell-vertex method [18, 35] and
offers third-order spatial and temporal accuracies. Variations in the filter size due to
non-uniform meshes are not directly accounted for in the LES models. Changes in cell
topologies and sizes are only accounted for through the use of the local cell volume, that
is 4 = V1/3
cell . Grid refinement needs therefore to be carefully controlled for the LES
model to operate efficiently. Such effects are beyond the scope of this work although
great care has been taken to minimize the consequences on the predictions.
3. Configuration
The flow geometries simulated are composed of a rectangular duct with a constant cross
section of 30.9 mm in height and 50 mm in width. The length of the duct is 230 mm
and air flows through the duct with a stream speed of 60 ms−1. The large Reynolds
number, Re = 150, 000 (based on the duct height), improves performance by reducing
the boundary-layer thickness and increasing the overall turbulence level. Two opposed
rows of five in-line JICF are placed 100 mm downstream of the inlet duct section. Fuel
† is injected perpendicularly to the transverse flow-through the JICF holes with an
injection speed of 195 ms−1. The jet-to-mainstream momentum flux ratio:
J =ρjetv
2jet
ρairv2air
, (10)
is 10.4 and the speed ratio:
R =vjet
vair
, (11)
equals 3.25. The injector’s geometry is incorporated in the LES and consists of a
circular duct perpendicular to the transverse flow with a diameter of 1.4 mm. All
these components constitute the first configuration simulated through LES and referred
to as Case 1.
† In practice for Bochum experiment, fuel is replaced by an acetone seeded air source. For LES this
secondary source is modeled by a passive scalar source with consistent fluid characteristics
LES of mixing enhancement 6
The second geometry designated by Case 2 consists of Case 1 with added mixing
devices. The devices located in the rectangular duct prior to the JICF injection planes
intend to generate large flow scales (with size of the order of the duct height). The
objective is to enhance fuel mixing through increased vorticity [36]. The geometry is
illustrated in figure 1. Note that experimental measurements are available for both cases
(a)
X
Y
Z(b)
X
Y
Z
Figure 1. Flow configurations: (a) Case 1, without mixing devices, (b) Case 2, with
mixing devices.
and were obtained in Bochum [16], Germany.
3.1. Grid characteristics and boundary conditions
The grid generated for the two computational domains is unstructured and composed
of tetrahedra. Limits in computational cost impose to control the number of grid points
and cells. The main difficulty is to adapt the mesh refinement to the flow scales for
good LES predictions. Injection areas must be sufficiently resolved to capture the
proper range of length scales. Note hat each jet injection tube is meshed to resolve
the flow upstream of the injection section. Table 1 specifies mesh parameters for the
two geometries: without mixing devices, Case 1, and with mixing devices, Case 2.
Calculations were conducted on 32 processors of a SGI Power Challenge (512 processors)
Nodes Cells Description
number number
Case 1 285 000 1 500 000 Baseline flow: 10 JICF
Case 2 375 000 2 000 000 Baseline flow + mixing devices
Table 1. Specificities of the two computational grids, Case 1 and Case 2.
and one convective time (based on the main canal length) took about 16 hours (wall
clock) for Case 1 and 22 hours (wall clock) for Case 2.
Boundary conditions need special care. A schematic representation is shown on
figure 2 and details of the conditions used in the LES is given in table 2. Inlet boundary
LES of mixing enhancement 7
(a)
(b)
Figure 2. Plot (a) illustrates the computational model and boundary conditions as
imposed in the LES of Case 1 and Case 2 (cf. table 2). Plot (b) shows details of the
mixing devices as used in Case 2.
conditions impose the mass flux in agreement with Bochum’s experiments. The NSCBC
method [25, 9, 37] is employed for both the inlet and the outlet in order to reduce
acoustic disturbances. Note that in the computational model, figure 2, the rectangular
duct terminates into a settling chamber to minimize the influence of the downstream
boundaries. For identical reasons, square chambers with 6 mm sides and 6 mm heights
are located upstream of the circular JICF ducts. The JICF duct geometry is circular,
1.4 mm in diameter (as imposed by the experimental setup) and 1 mm in height. The
inlet bulk velocity imposed at these injection chambers is estimated at 8.34 ms−1 to
yield an injection bulk velocity of 195 ms−1 in the main chamber (as specified by the
LES of mixing enhancement 8
experiment). Figure 2(b) presents the mixing devices for Case 2: boundary conditions
(except inlets and outlets) are no-slip adiabatic walls.
Patch # BC Imposed quantities
1 Air inlet velocity, temperature
2 Injection inlet velocity, temperature
3 Outlet pressure
4 No-slip, adiabatic wall stream-wise velocity component
Table 2. LES set of boundary conditions.
The use of settling chambers down-stream of the main duct and upstream of the jet
injection system results in a computational domain which differs from the experimental
setup. The potential drawback of such an approach may appear if acoustics plays a
significant role in determining the flow solution. Indeed the modified inlet and outlet
impedances will result in different acoustic charcteristic times henceforth modifying
the mixing mechanisms. For the case under investigation acoustics is believed to be
secondary and settling chambers are used to facilitate the treatment of the acoustic
field at the boundaries. The intend is to minimize the effect of the acoustic noise due
to an approximate treatment of the boundary conditions so as to minimize its impact
on the LES results.
3.2. Initial conditions
In order to smoothly converge toward fully established LES results a particular
methodology is assembled to construct meaningful initial conditions. The aim of the
approach is to minimize computational efforts while suppressing undesired behaviors
generated by the inadequate initialization. Two steps are taken: (a) determination of
an approached solution on a coarse grid and for which the initial artifacts due to the
initial guess are removed; (b) use of the converged solution obtained in (a) as the initial
guess for the final LES. Both steps require advancement in time of approximately one to
two flow-through times before disappearance of the initial non-physical response. The
computational grid used in (a) is roughly twice as coarse as the final LES mesh and
allows a non-negligible gain in computing efforts (AVBP was measured to be 5 to 6
times faster on grid (a) than on grid (b)). A flow-through time was run on grid (b)
before analyzing LES results. Average quantities were measured over a time span of
two flow-through time.
4. Results
Predictions of the LES are compared to existing measurements for both configurations
and prior works on JICF. First, the flow topology obtained with LES is analyzed. The
changes in the flow behavior implied by the changes in geometry are illustrated for
LES of mixing enhancement 9
the velocity field and the fuel concentration. Second, a quantitative analysis of the
various phenomena is presented through the use of statistical diagnostics. These include
presentation of one point Probability density functions (Pdf’s) and indices characterizing
temporal and spatial mixing.
4.1. Flow topology
JICF have been heavily investigated due to their multiple industrial implications [38]
and experimental observations go back to the thirties with [39]. The systematic analysis
of the JICF started in the seventies with the discovery and acceptance of coherent
structures [40, 41, 42]. Four dominant large scale motions are believed to be determinant
in the JICF (figure 3). The prominent vortex system is the Counter-rotating Vortex
Pair (CVP). The three other vortices, the jet shear layer vortices, the wake vortices
and the horseshoe vortex play a lesser role in the far field of the jet. Based on past
Figure 3. Vortex system in a JICF (from Fric & Roshko [43]).
results [44, 45] the momentum flux ratio, J , defined in (10), emerges as determining the
JICF flow features. In addition to J one notes the importance of the jet injection hole
shape and size [46, 47], the jet inlet velocity profile [48], the initial jet penetration angle
[49] and the presence of other jets [50] . Numerical simulations have been performed
for the JICF in the eighties with [41, 51, 52, 53, 54, 55, 56] and more recently with
[57, 58, 59, 60, 61].
Downstream of the injection holes instantaneous and averaged LES fields allow
the identification of the different vortex systems which control the entrainment and the
mixing of the fuel. For Case 1 the flow topology can be described using observations
made for simple JICF: jet trajectory and main coherent structures such as Counter-
rotating Vortex Pair (CVP) are featured. For Case 2, CVP can also be observed but is
LES of mixing enhancement 10
dominated by a large structure developing downstream of the mixing region. Averaged
motion and instantaneous scalar fields are also shown.
4.1.1. Large scale motions Jet trajectory visualization of the opposed JICF is
illustrated in figure 4 for Case 1. Both pictures feature a streamwise plane located
in the middle of the spanwise direction (i.e. going through z = 25 mm). The averaged
fields of fuel concentration and vorticity are represented. Based on these quantities, the
jet is seen to bend in the direction of the cross-flow just after exiting the injection nozzle.
In a JICF two mechanisms explain this deflection [62]: the first mechanism is induced by
the pressure gradient between the upstream (high pressure) and the downstream (low
pressure) flow over the wall at the jet exit. The second mechanism is the entrainment of
the jet flow by the cross-flow stream. On figure 4 the near wall injection region is zoomed
to characterize the lean fuel zone and large levels of vorticity (cf. zoomed frames).
Figure 4. Case 1: Jet trajectories visualization in a streamwise plane passing
through, z = 25mm: averaged fields of, top, fuel concentration, and bottom, vorticity.
The CVP topology is shown in figure 5. The pressure difference provides the
force that deforms the jet and contributes to the development of this JICF prominent
structure. Figure 5 shows iso-surface of the Q-criterion [63] colored by the streamwise
LES of mixing enhancement 11
component of the vorticity vector. The iso-surface level is taken at 2 107s−2. For each
JICF, two vortices of opposite signs (grey, negative values and blue, positive values)
are clearly observed and structures subsist over the streamwise direction. Three planes
located downstream of the injection point at the axial positions, x = 5, 20 and 42 mm,
show the scalar fuel concentration (dark values suppose large concentration of fuel).
Successive phenomena appearing over the canal length are clearly identified. The CVP’s
rotate around each other, interact with the other jet CVP’s and finally merge.
Figure 5. Case 1: instantaneous illustration of the counter-rotating vortex pairs
(iso-surface of the Q-criterion colored by the streamwise component of the vorticity
vector) and fuel concentration in three planes located at 5, 20 and 42mm downstream
of the injection point.
Figure 6 displays the averaged velocity field in planes located at 10, 20, 30, 40, 50
and 70 mm downstream of the injection point. The CVP’s evolution over the canal
length is clearly observed.
The forthcoming results are devoted to the coherent structures developing
downstream of the mixing devices, Case 2. The JICF coherent structures are still
LES of mixing enhancement 12
Figure 6. Case 1: modulus of averaged fields of velocity (v2 + w2)1/2 in an x-plane
located at 10, 20, 30, 40, 50 and 70 mm downstream of the injection point.
present, but each jet trajectory is influenced by the organized motion in the mainstream,
figure 7.
In figure 8, cross-stream planes located at the streamwise locations −2,−1, 1, 4, 8
and 12 cm allow visualization of the organized motion. Close to the mixing device the
four distinct zones on the left hand side of Plane 1 correspond to a single coherent
structure. The same structure is found on the right hand side of Plane 1. It is clearly
observed in Planes 2, 3, 4, 5 and 6 that these coherent structures consist of two large
rotating structures whose intensity decreases slowly in the far field region. From the LES
predictions, the two vortical structures rotate clockwise. Figure 9 and figure 10 show
the mixing device influence on the JICF’s near field. They present the instantaneous
field of scalar fuel concentration in different sections. The organized motion present in
the mainstream induces faster interaction between each jet when compared to Case 1.
Two of the ten jets end up blocked against the wall (the top-right jet and the down-left
jet). The threshold value used in figure 9 and 10 for the fuel mass fraction is fixed at
0.17; the dark values represent maximum of concentration. In figure 9, planes 1, 2, 3
and 4 are respectively situated at x = 2, 5, 8 and 10mm downstream the injection point.
On Plane 1, CVP of the ten JICF are clearly seen. On Plane 2 (5 mm downstream the
injection point), the pairing mechanism appears and gets clearer at x = 8 and 10 mm
(Planes 3 and 4). In figure 10, Planes 5 and 6 are situated at Y = 3 (near the down
wall) and Y = 15.45mm (middle of the channel). As observed previously, two jets stand
against the wall over a good portion of the duct, see Plane 6 of figure 10.
LES of mixing enhancement 13
Figure 7. Case 2: instantaneous illustration of the counter-rotating vortex pairs
(iso-surface of the Q-criterion colored by the streamwise component of the vorticity
vector) and fuel concentration in three planes located at 5, 20 and 42mm downstream
of the injection point.
4.1.2. Implications on mixing To understand the role of the mixing devices, it is
convenient to visualize averaged fields of the mixing index, Yf ×Yo. With this definition
and in the configuration studied, non-zero values of Yf×Yo identify regions where mixing
is taking place while zero value correspond to regions with only fuel or air. Figure 11(a)
for Case 1 and figure 11(b) for Case 2 show spatial distributions of Yf×Yo as obtained
from the LES. Both plots display vertical planes located at axial positions, x = 5, 20
and 50 mm (Planes 1, 2 and 3 respectively) downstream of the injection point. The
threshold value is so that Yf×Yo = 0.1. For Case 1, figure 11(a) shows ten distinct CVP
regularly spaced which develop over the canal length, Plane 1 and 2. At x = 50 mm,
Plane 3, the merging mechanism is complete. Note that at this stage the respective
opposed jets have not interacted yet. As expected for this configuration mixing occurs
essentially within the CVP’s. Figure 11(b) shows the second configuration (Case 2)
LES of mixing enhancement 14
Figure 8. Case 2: modulus of averaged fields of velocity (v2 + w2)1/2 in a x-Plane
located at 2 and 1 cm upstream of the injection point (Planes 1 and 2) and at 1, 4, 8
and 12 cm downstream of the injection point (Planes 3, 4, 5 and 6).
which exhibits significant differences from the mixing point of view. The two rotating
structures deviate each jet trajectory and homogeneisation is enhanced. On Plane 3,
mixing is lower and more uniformly distributed in comparison to Case 1. Peaks in
mixing occur in opposite corners (top right and bottom right corners) which correspond
to the trapped JICF illustrated in figure 10. Addition of the mixing devices clearly
enhances mixing and results in a homogeneous mixture sooner than in Case 1 because
of the enhanced stirring.
4.2. Statistical Analysis
Statistical analysis of the LES predictions is performed to characterize the mixing
mechanisms presented in section 4.1. All results are obtained through temporal and
spatial averages of the LES simulations. From these operations indices and Pdf’s are
constructed and presented in this section. When available unsteady RANS results
obtained by Siemens, PG [17], and measurements obtained by the University of Bochum
[17] are added for comparisons.
4.2.1. Mixing indices Two mixing indices, the Spatial Mixing Deficiency (SMD)
and the Temporal Mixing Deficiency (TMD), are investigated in planes located at
x = 46, 82, 160 and 228 mm downstream of the injection point. The SMD and
TMD are indices based upon instantaneous measurements of the fluorescence intensity,
LES of mixing enhancement 15
Figure 9. Case 2: Scalar fuel concentration in planes located at x = 2, 5, 8 and
10 mm. The threshold value is 0.17 and dark grey values represent zones where the
scalar fuel concentration is larger than 0.17.
Ii,k, as obtained in the experiment (the technique used is the Planar-Laser Induced
Fluorescence, PLIF) and measures the local fuel mass fraction. In the notation, Ii,k, the
subscript i refers to the position, xi, while k points to the time instant k. Averages over
a series of n images (n times), the mean and root mean square, RMSi, are deduced by:
< Ii >=1
n
n∑
k=1
Ii,k, and RMSi =
√√√√ 1
n− 1
n∑
k=1
(< Ii > −Ii,k)2. (12)
The SMD index corresponds to a planar average and measures the spatial heterogeneity
of the mixture (a zero SMD value indicates perfect mixing in this plane):
SMD =RMSplane(< Ii >)
Avgplane(< Ii >), (13)
where,
Avgplane(< Ii >) =1
m
m∑
i=1
< Ii >, (14)
and,
RMSplane(< Ii >) =
√√√√ 1
m− 1
m∑
i=1
(< Ii > −Avgplane(< Ii >))2. (15)
LES of mixing enhancement 16
Figure 10. Case 2: Scalar fuel concentration in planes located at y = 15.45 and
3 mm (top and bottom plot respectively). The threshold value is 0.17 and dark grey
values represent zones where the scalar fuel concentration is larger than 0.17.
(a) (b)
Figure 11. The mixing index Yf × Yo in vertical planes located at the streamwise
positions x = 5, 20 and 50 mm downstream of injection point. Plot (a) corresponds to
Case 1 and plot (b) to Case 2.
LES of mixing enhancement 17
The TMD index is given by a planar average of the temporal fluctuations and measures
the temporal heterogeneity of the mixture:
TMD = Avgplane(RMSi
< Ii >). (16)
LES estimates of TMD and SMD are obtained by considering that the fuel mass
fraction Yfi, at point i, is directly proportional to the fluorescence intensity, Ii, measured
in the experiments. It is however necessary to take into account two major factors for
a direct comparison between experiments and LES results:
• The laser diagnostic techniques used in these experiments are realized with 5200
measurements and data is filtered. For the LES predictions, interpolation routines
are used to yield values in the planes of interest when they do not correspond
directly to grid points. The resulting values depend on the mesh precision and
decreases from 4008 data points for Plane 1 to 346 data points for Plane 4 (outlet
of the chamber).
• Temporal averages are obtained with high precision for the LES predictions because
time-steps are smaller than the typical acquisition time step in the experiment.
The temporal resolution is therefore higher in LES than in the experimental set up.
However only filtered quantities are accessible in LES contrarily to experimental
measurements which are exact (before treatment). It results that RMS’s obtained
by LES may under-estimate experimental measurements.
Figure 12 illustrates estimates for the indices obtained from LES predictions and
unsteady RANS ‡ results [17]. The two indices, SMD and TMD are presented as a
function of downstream locations. Estimates of the SMD are illustrated on figure 12(a).
For Case 1, the LES results show a similar decay rate than the experimental values
near the exit of the main duct. Closer to the injection point the decay rate is clearly
over-estimated. Two reasons can explain this difference: first, as mentioned above,
each diagnostic uses filtering techniques (for experiments) or interpolations (for LES
data). The second reason is that the small flow scales need to be well resolved. Despite
the mesh refinement, the grid may still be too coarse in the near field JICF region
for Case 1. Finally, SGS terms are not accounted for in the numerical/experimental
comparison which constitutes another source of error particularly in the near injection
field. For Case 2, LES predictions are in good agreement with experimental values.
Although the mesh refinement in the neighborhood of the JICF is the same for both
geometries, the difference found between the experimental data and LES results of
Case 1 is not seen for Case 2. For that latter case, the flow topology is characterized
by very large structures which are well resolved by LES. Estimates of the TMD are
shown on figure 12(b). Similarly to the SMD, LES predictions for Case 1 over-predict
the experimental measurements (10% higher). For Case 2, LES predictions are much
closer to the experimental values. The improvement due to the LES approach over the
‡ Note that the RANS predictions are obtained with adjusted model coefficients to yield descent
predictions.
LES of mixing enhancement 18
(a) (b)
Figure 12. Comparison of, (a) the Spatial Mixing Deficiency and (b) the Temporal
Mixing Deficiency.
RANS methodology is clearly illustrated as long as the mesh refinement is properly
tuned.
4.2.2. Probability density functions The probability density function at point xi follows
[64, 65, 66, 67]:
f(ξ; xi) ≡ {Probability that the event αik = ξ occurs}, (17)
where ξ is the statistical representation of α and xi the point under observation. Each
function, f(ξ; xi), is constructed from the time series αik (i kept fixed for a given xi) so
that mean temporal quantities are obtained through integration:
< αi > =
∫ +∞
−∞
ξ f(ξ; xi) dξ,
RMSi =
∫ +∞
−∞
(ξ− < αi >)2 f(ξ; xi) dξ (18)
Based on definition (17), only temporal information relative to the point xi can be
extracted. Spatial information can only be inferred through the comparison of the Pdf’s
obtained at different locations (i.e.: different values of xi). For clarity and simplicity
the Pdf of the jet air scalar concetration Yf is not directly used. Instead temporal
homogeneity of the flow is measured by the parameter:
α = 100
(Yf
Yo
)local(
Yf
Yo
)global
. (19)
In (19) global quantities are determined from the air and fuel mass fluxes. When
α = 100 homogeneity is reached, while α > 100 and α < 100 correspond respectively to
LES of mixing enhancement 19
fuel rich and lean realizations. The shape of the Pdf is a strong indicator of the temporal
evolution taking place at a given location. Spatial variability of the shape of the Pdf is
also useful to characterize the mixing process taking place at the various locations. To
quantify these variations, four points in a downstream plane located 228 mm from the
injection points are probed to construct the respective Pdf’s (figure 13(a)). Similarly
the evolution of the Pdf along the centerline of the computational domain is illustrated
for the points indicated on figure 13(b).
(a) (b)
Figure 13. Location of the points for which Pdf’s are constructed, (a) position of the
first four points in the cross-stream plane located 228 mm downstream the injection
points, (b) position of the remaining points in the domain.
Figures 14 & 15 for Case 1 and Case 2 depict the spatial variability of the temporal
homogeneity in plane x = 228mm (near the exit). With no mixing device, the Pdf at
points 1 and 2, figure 14(a) &(b), differ slightly. The mean temporal homogeneity
tends toward α ≈ 135 % and corresponds to fuel rich regions. Variations from one
jet to the next are for that case quite limited and the underlying temporal mixing
processes taking place at these points can be considered equivalent. Looking at Pdf’s
at points 3 and 4 (i.e. figure 14(c) &(d)), the distributions are similar. The means are
respectively 175 % and 115 % for points 3 and 4 indicating a richer region at point 3.
It is important to note that for this configuration the Pdf’s shape approaches the delta
function distribution indicating that the rate of mixing mechanism is small. All points 1
to 4 are usually associated with fuel rich regions which supposes the existence of strong
spatial inhomogeneities and almost no mixing of the jet fluid.
Introduction of the mixing devices, figure 15, results in more doubled peaked Pdf’s
(cf. figure 15). All distributions are centered around α ≈ 110 % which indicates that
full mixing is almost reached in time and space. Temporal homogeneity is however not
guaranteed at all points. Indeed, fuel rich pockets can be found when the device is
used (cf. peaks on the right hand side of figure 15). The probability for such events is
nonetheless quite low in comparison with the peak at α = 110 %. Anyhow mixing in
time as well as in space is greatly improved when compared to Case 1.
LES of mixing enhancement 20
(a) (b)
(c) (d)
Figure 14. Pdf’s at the downstream positions for Case 1. Plots (a), (b), (c) and (d)
correspond respectively to points 1, 2, 3 and 4 (cf. figure 13(a)).
(a) (b)
(c) (d)
Figure 15. Pdf’s at the downstream positions for Case 2. Plots (a), (b), (c) and (d)
correspond respectively to points 1, 2, 3 and 4 (cf. figure 13(a)).
LES of mixing enhancement 21
(a) (b)
(c) (d)
Figure 16. Pdf’s at the downstream positions for Case 1. Plots (a), (b), (c) and (d)
correspond respectively to points 5, 6, 7 and 1 (cf. figure 13(b)).
(a) (b)
(c) (d)
Figure 17. Pdf’s at the downstream positions for Case 2. Plots (a), (b), (c) and (d)
correspond respectively to points 5, 6, 7 and 1 (cf. figure 13(b)).
LES of mixing enhancement 22
Looking at the Pdf’s evolution along the duct centerline, figures 16 & 17, one
can infer information about the spatial evolution of the mixing process taking place
in the downstream direction. Figure 16 shows the Pdf’s obtained for points 5, 6, 7
and 1 (cf. figure 13) for Case 1 while figure 16 refers to results obtained at the same
locations but for Case 2. Concentrating on Case 1 and in the near field region of
the jet, figure 16(a), the mean temporal homogeneity is estimated at α ≈ 65 % which
corresponds to a fuel lean region. Peaks at α = 25, 50 and 75% indicate the tendency to
have alternatively fuel rich pockets probably issued from the nearby JICF’s. Traveling
further downstream, point 6 (cf. figure 16(b)), the distribution is quite different. First,
temporal homogeneity at this point is almost achieved (at least in the mean sense):
the Pdf is centered around the value of 100 %. Second, temporal homogeneity is not
guaranteed instantaneously. The distribution is symmetric with two peaks respectively
located at α = 50 and 150 %. Such a shape is indicative of large fuel rich and lean
structures passing through the point. This could be explained for example by the
possible flapping of the jet at this location. Further downstream at points 6 and 1
(cf. figure 16(c) &(d)), although temporal homogeneity is almost achieved in the mean
sense, full mixing is clearly not ensured instantaneously: the distributions tend toward
delta functions centered at α ≈ 130 % for point 1 which corresponds to a highly rich
region.
Introduction of the mixing device improves the mixing taking place along the
centerline. At points 5 and 6, figure 17(a) &(b), the Pdf’s take on a Gaussian like
shape centered respectively around α = 125 and 110 %. Such shaped Pdf’s allow to
state that the mixing is essentially governed through highly turbulent processes. Their
efficiency is linked to the diminishing variance of the Pdf’s. Going further downstream,
points 7 and 1, the Pdf’s take on similar shapes and approach a double peaked function.
This demonstrates that full mixing is not guaranteed instantaneously for these points.
Indeed, fuel rich regions may be encountered occasionally (peaks at α = 150 %) and
prove that large scale motions play a determining role in ensuring full mixiness prior
the duct exit.
5. Conclusions
In this work initiated by Siemens PG, Germany, the design of the fuel injection system
is addressed using LES. The aim of the study is to demonstrate the ability of this new
computational approach to assess two design options for the proper mixing of fuel and
air prior to its combustion. The quality of the design is gauged based on the temporal
and spatial homogeneity of the mixture at the exit of the chamber. The importance of
such criteria is revealed in the fact that fuel inhomogeneities entering the combustion
zone may potentially trigger combustion instabilities whose consequences are: increase
of the pollutant formation and the potential damage of the entire gas turbine engine.
Assessing the mixing quality of a design is therefore essential for the next generation of
gas turbines.
LES of mixing enhancement 23
LES results are presented for two design options based on the simpler Jet In Cross-
Flow (JICF) configuration. The first geometry consists of ten opposed JICF’s issuing
fuel into air flowing in a rectangular duct. The second design is based on the first
one with an added mixing device. Instantaneous LES results clearly illustrate major
differences in the flow topologies. While the first design follows the known evolutions
of interacting JICF’s, the introduction of the mixing device generates a flow dominated
by two rotating structures scaling with the hight of the duct. These rollers increase the
entrainment of the JICF’s and the strength of the mixing process when compared to
the first design. Statistical analyses reveal that the Spatial Mixing Deficiency (SMD)
and Temporal Mixing Deficiency (TMD) indices are appropriately estimated by the LES
predictions when compared to the measurements performed by the University of Bochum
[17]. Finally the Probability density function (Pdf) analysis of the LES clearly assesses
the predictions in term of temporal and spatial homogeneity of the mixture at various
locations in the computational domain. Full temporal and spatial homogeneity at the
exit of the test section is not achieved by any of the two design options. However addition
of the mixing device greatly improves the results obtained from the LES predictions and
outperforms results obtained for the first design.
LES of mixing enhancement 24
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LES of mixing enhancement 27
Figure captions
Figure 1. Flow configurations: (a) Case 1, without mixing devices, (b) Case 2, with
mixing devices.
Figure 2. Plot (a) illustrates the computational model and boundary conditions as
imposed in the LES of Case 1 and Case 2 (cf. table 2). Plot (b) shows details of the
mixing devices as used in Case 2.
Figure 3. Vortex system in a JICF (from Fric & Roshko [43]).
Figure 4. Case 1: Jet trajectories visualization in a streamwise plane passing through,
z = 25 mm: averaged fields of, top, fuel concentration, and bottom, vorticity.
Figure 5. Case 1: instantaneous streamwise component of the vorticity vector (iso-
surfaces level of 2.5 104s−1) and fuel concentration in three planes located at 5, 20 and
42 mm downstream of the injection point.
Figure 6. Case 1: modulus of averaged fields of velocity (v2 + w2)1/2 in an x-plane
located at 10, 20, 30, 40, 50 and 70 mm downstream of the injection point.
Figure 7. Case 2: instantaneous streamwise component of the vorticity vector (iso-
surfaces level of 2.5 104s−1) and fuel concentration in three planes located at 5, 20 and
42 mm downstream of the injection point.
Figure 8. Case 2: instantaneous streaklines visualization, (light grey for fuel and
dark grey for main flow air).
Figure 9. Case 2: modulus of averaged fields of velocity (v2 + w2)1/2 in a x-Plane
located at 2 and 1 cm upstream of the injection point (Planes 1 and 2) and at 1, 4, 8
and 12 cm downstream of the injection point (Planes 3, 4, 5 and 6).
Figure 10. Case 2: Scalar fuel concentration in planes located at x = 2, 5, 8 and
10 mm. The threshold value is 0.17 and dark grey values represent zones where the
scalar fuel concentration is larger than 0.17.
Figure 11. Case 2: Scalar fuel concentration in planes located at y = 3 and 15.45mm.
The threshold value is 0.17 and dark grey values represent zones where the scalar fuel
concentration is larger than 0.17.
Figure 12. The mixing index Yf × Yo in vertical planes located at the streamwise
positions x = 5, 20 and 50 mm downstream of injection point. Plot (a) corresponds to
LES of mixing enhancement 28
Case 1 and plot (b) to Case 2.
Figure 13. Comparison of, (a) the Spatial Mixing Deficiency and (b) the Tempo-
ral Mixing Deficiency.
Figure 14. Location of the points for which Pdf’s are constructed, (a) position of
the first four points in the cross-stream plane located 228 mm downstream the injection
points, (b) position of the remaining points in the domain.
Figure 15. Pdf’s at the downstream positions for Case 1. Plots (a), (b), (c) and
(d) correspond respectively to points 1, 2, 3 and 4 (cf. figure 13(a)).
Figure 16. Pdf’s at the downstream positions for Case 2. Plots (a), (b), (c) and
(d) correspond respectively to points 1, 2, 3 and 4 (cf. figure 13(a)).
Figure 17. Pdf’s at the downstream positions for Case 1. Plots (a), (b), (c) and
(d) correspond respectively to points 5, 6, 7 and 1 (cf. figure 13(b)).
Figure 18. Pdf’s at the downstream positions for Case 2. Plots (a), (b), (c) and
(d) correspond respectively to points 5, 6, 7 and 1 (cf. figure 13(b)).
LES of mixing enhancement 29
Table captions
Table 1. Specificities of the two computational grids, Case 1 and Case 2.
Table 2. LES set of boundary conditions.
Recommended